Hex Grids

January 25, 2015

Solid state materials pack together in a myriad of ways. The crystal structures are typically referred to with the name of the archetype compound. This ranges from fairly common English words such as ‘diamond’ and ‘rock-salt’ to, ‘zinc-blend’ ‘wurtzite’ and, of course, ‘perovskite’.

The atomic packing within these structures is fully described by their space-group, of which there are 230 to choose from. These space groups take the crystallographic (periodic structure) compatible point groups, and combine them with different possible lattice vectors.

Some structures are fiendishly complicated, others are effectively cubic grids (perhaps with lattice vectors which are neither orthogonal nor equal in length). Even very complicated structures often have a sub lattice which packs cubically, or with hexagonal close packing.

One of the perhaps surprising things is that hexagonal packing and cubic packing are extremely similar to one other – in particular cubic (BCC and FCC) and hexagonal close packing both have a coordination number of 12. Each sphere (atom) has 12 nearest neighbours.

This is very useful, when you are building a computer model for a large structure. You can store the data about site occupancy in a simple array[][][], and calculate various necessary metrics (in particular real space distance vectors) via simple (i.e. free, in terms of computer time) arithmetic.

I came across an article on hexagonal grids today; much of it was startling familiar, but in that way that it was giving you a framework and lexicon for understanding for what you’d hacked together with a partial glimpse of true understanding.

(It’s written from the perspective of a games developer, but it’s directly applicable to scientific computation – just don’t tell the Research Councils how similar our austere science is to a game engine!)